Prior art hardware and software for graphic modeling systems are predominantly designed to store, manipulate, and render models that are represented by polygons, e.g., triangles. Despite technical advances, hardware and software continue to have limits in the number of polygons they can store and process interactively. These limits depend on the requirements of each specific application. Applications that require quality images, such as movies and medicine, may require a large number of polygons, while other applications, for example, those designed for portable devices, are constrained to using a small number of polygons due to processing and memory limitations. Applications that require views of the same model at varying distances, for example, games, can require any number of polygons, depending on their position in a scene.
A number of methods are known for transforming complex objects into level-of-detail (LOD) models having various resolutions. Lower resolution models are simpler expressions of the input model. LOD models represent the input object with varying accuracy and geometrical complexity. LOD methods typically process a subset of the available geometrical representations, e.g., points, polygons, Bezier patches, non-uniform rational B-splines (NURBS), volumes, implicit surfaces, and CSG models, and usually generate a triangle mesh that can be rendered by standard rendering engines.
Several of the known methods perform a sequence of transformations which provide various LOD representations of the model, including an initial representation, intermediate representations, and a final output mesh of triangles. LOD models should allow smooth interpolation between representations at differing resolutions. However, many of these methods exhibit discontinuities during interpolation, and perform poorly, if at all, at lower resolutions due to compounded errors caused by repeated transformations.
Many three dimensional models are acquired by using techniques such as laser range scanning, or imaging techniques, for example, magnetic resonance imaging (MRI) and computerized tomography (CT), to generate models with a high level of detail. Representing these highly detailed models may require the system to generate millions of triangles. To then render these triangles in real-time, as required by games and medical applications, is impractical with current hardware and software methods.
Automatic modeling methods are known that generate meshes of given sizes (resolution) from the model's input geometry, usually polygons. For a survey of available methods see Garland, “Multiresolution modeling: Survey & future opportunities,” Eurographics State of the Art Reports, pp. 111-131, 1999, and Heckbert et al. “Survey of polygonal surface simplification algorithms,” Multiresolution Surface Modeling Course, SIGGRAPH '97, 1997.
Because arbitrary polygons can always be decomposed into triangles, a simple mesh consists only of triangles. Garland notes that modeling systems that simplify the model can be split into two types: refinement and decimation methods. Refinement methods take a simplified representation of the model's geometry, and add to the simplified representation where it is inadequate. Decimation methods are the conceptual opposite of refinement. Elements of the model are iteratively removed until the desired level of resolution is reached.
Due to better performance, refinement models tend to be the more popular choice in applications such as curve approximation. However, in order to apply the refinement methods to an original polygonal model, it is necessary to have a base mesh, which is a simplification of the original model, yet retains the exact same topology. Generating the base mesh can be complicated, leading to the use of decimation methods.
Vertex clustering, as described by Rossignac et al. in “Multi-resolution 3D approximations for rendering complex scenes,” Modeling in Computer Graphics: Methods and Applications, pp. 455-465, 1993, is one decimation method that quickly simplifies mesh models. In its simplest form, vertex clustering partitions the initial geometry into cells for a given level of detail, and clusters all the vertices that lie within each cell together. New triangles are formed by replacing each vertex in the original mesh with its cluster in the new mesh. Degenerate triangles are removed where two or more of the vertices of an input triangle map to the same cluster. Although vertex clustering is fast, no attempt is made to preserve topology. This can lead to poor quality approximations of the simplified model, especially when the final number of polygons is much smaller compared to the number of polygons in the input mesh. An additional disadvantage of the vertex clustering method is that the entire decimation process must be restarted for a different resolution.
Several similar techniques avoid the need for restarting by iteratively performing an operation on the polygon mesh until the desired level of simplification has been reached, for example see Heckbert et al. “Optimal triangulation and quadric-based surface simplification, Computational Geometry, 14:49-65, 1999, Hoppe et al. “Mesh optimization,” Proceedings of SIGGRAPH '96, pp. 19-26, 1996, Lindstrom et al. “Fast and memory efficient polygonal simplification,” Proceedings of IEEE Visualization '98, pp. 279-286, 1998, and Schroeder et al. “Decimation of triangle meshes,” Proceedings of SIGGRAPH '92, pp. 65-70, 1992.
These methods generate a progression of polygon meshes, permitting different resolutions to be generated without starting from scratch. At each stage of the simplification process, the choice of which polygons to decimate depends on the amount of error that can be tolerated. The choice of the error metric leads to a trade-off between accuracy and efficiency. Accurate error metrics compare the approximated model with the original model to provide better quality meshes, at the cost of increased processing time, see Hoppe et al., “Mesh optimization,” Proceedings of SIGGRAPH '96, pp. 19-26, 1996, Popovic et al., “Progressive Simplical Complexes,” Proceedings of SIGGRAPH '97, pp. 217-224, 1997. Efficient error metrics, such as a quadric error metric, provides very fast decimation, but compound any errors during each iteration, see Garland et al. “Surface simplification using quadric error metrics,” Proceedings of SIGGRAPH '99, pp. 209-216, 1999.
Schroeder et al. iteratively removes vertices from the input mesh, one at a time. Damaged triangles are removed, and resultant holes are re-triangulated after projecting a local surface onto a plane. This introduces a restriction to manifold surfaces.
Other methods focus on vertex pair contraction. There, pairs of vertices are combined. The methods Garland et al., Lindstrom et al., and Hoppe are examples of vertex pair contraction. Edge contraction methods only combine the vertices connected by an edge, and remove the triangles that share the same edge. These methods preserve the topology of the model in the approximations. The more general case of vertex pair contractions, where the pair of vertices does not have to be connected, and make no effort to preserve topology. Although this can be beneficial, it can sometimes lead to poor decisions because arbitrary parts of a model are stitched together.
Gopi et al., in “Simplifying spline models,” Computational Geometry, 14:67-90, 1999, describe a technique for generating level-of-detail models composed of triangular Bezier patches. These curved level-of-detail (C-LOD) models can be generated from input spline patches. Their technique can also apply a series of transformations to the existing geometry, e.g., diagonal swapping, vertex removal, or patch merging. However, that method is limited to working with spline patches, although the resulting Bezier patches can be converted to triangles, either statically or dynamically.
All of the methods described above, except for vertex clustering, work by iteratively applying one or more transformations to the input mesh to generate one or more level-of-detail meshes. The methods are limited to using the geometry of the input mesh to measure the error introduced by the simplification process. The methods can also be sensitive to the original discretization of the geometry, i.e., input vertex placement.
Due to the refining nature of these methods, the time required to generate an approximation is dependent upon the size of the input mesh. For example, Hoppe's method requires about an hour to simplify a model with 70,000 polygons, and would require a significantly longer time for larger models. Although Garland's method is significantly faster, it produces low quality LOD models due to compounded error. None of the prior art approaches are practical for real-time applications, especially when the models are comprised of millions of polygons, as is the case for scanned models. Furthermore, the prior art methods produce low quality models when the polygon count is low, as is required for games.
Therefore, there is a need for a modeling system that can work on arbitrary input data, that does not compound errors, that can process models in real-time, that produces good models when the polygon budget is low, that is independent of the input vertex placement, and that can be used in conjunction with existing decimation methods.